Portfolio Policies with Unobservable
Markov Modulated Drift Process and Bounded Expected Loss.
J. Saß and R. Wunderlich
In: C. Fernandes, H. Schmidli, N.Kolev (eds.): Proceedings of the
Third Brazilian Conference: on Statistical Modelling in Insurance
and Finance, Maresias, March 25-30, 2007,
Institute of Mathematics and Statistics, University of Sao Paulo, 242-247, 2007.
Abstract :
In this paper we consider the computation of optimal portfolio
strategies for utility maximizing investors under a shortfall risk
constraint based on a financial market model with partial
information on the Markov modulated drift process. Without a risk
constraint the distribution of the optimal terminal wealth often is
quite skew and there are high probabilities for values of the
terminal wealth falling short a prescribed benchmark. This is an
undesirable and unacceptable property e.g. from the viewpoint of a
pension fund manager. Since a strict restriction to
portfolio values above a benchmark leads to a considerable decrease in
the optimal portfolio's expected utility, it seems to be reasonable to allow
shortfall and to restrict only some shortfall risk measure.
Value at risk (VaR) is a widely used risk measure and takes into
account the probability of a shortfall but not the actual size of
the loss. Therefore we use the so-called expected loss criterion
resulting from averaging the magnitude of the losses. Basak and
Shapiro (2001) show in Basak and Shapiro (2001) that the
distribution of the resulting optimal terminal wealth has more
desirable properties.
We use a
hidden Markov model (HMM) where the drift follows a continuous time
Markov chain. Sass and Haussmann (2004) show in that on market data utility maximization strategies
based on such a model can outperform strategies based on the
assumption of a constant drift parameter. Extending these results
we obtain quite
explicit representations for the form of the optimal terminal
wealth and the trading strategies which we compute by using Monte
Carlo methods. The same model was used in Gabih et al. (2005). Here we use a different class of risk
constraints, consider non-constant interest rates
and can give more explicit results on the existence and uniqueness
of an optimal solution.